where \(v_2(r;R)\)  is the union volume of two \(d\) -dimensional spheres of radius R, which is given for \(d=1,2\)  and 3 in Ref. [1]}. Using the union volume formulae in Eq. (REF ), we find that the two-point probability function for the first three dimensions has the form
\(S_2^{(1)}(r) = \exp \left[\ln \phi _1 h(r;\langle R \rangle ) \right]\) 
